The Spectral Window: Horizon Entropy and PBH Dark Matter

Thermodynamic consistency implies that modified horizon entropy changes black hole evaporation. Constraints on primordial black holes (PBHs) as dark matter assume the Hawking area law, S ∝ A, and corresponding temperature. Using Barrow entropy, S ∝ A^(1+δ/2), and deriving the thermodynamically consistent temperature T = c²(∂S/∂M)⁻¹ from the First Law, we show that for δ = 0.05 the evaporation spectrum shifts to lower energies, pushing the photon peak below the Fermi-LAT threshold for PBH-scale masses. PBH gamma-ray limits are not power-law sensitive to temperature they are exponentially sensitive once the thermal peak drops below the LAT band, creating a "spectral window" where PBHs can hide. The dominant effect is a sub-threshold spectral shift: once the peak falls below the LAT energy window, the IGRB constraint is set by the exponentially suppressed Wien tail rather than the thermal peak, weakening bounds by factors of 10⁴ to beyond 10¹⁶ depending on mass. This reopens a window M ≈ 1.8 × 10¹⁴–1.8 × 10¹⁵ g where PBHs can constitute all dark matter. The Barrow-to-Hawking f_PBH ratio is model-independent: IGRB normalization, halo profile, greybody factors, and secondary spectra cancel; only thermodynamics survives. Validation against BlackHawk v2.3, including exact greybody factors and PYTHIA secondary spectra, reproduces standard results at the ~6% level. We test the window against two critical challenges. First, MeV-band data from COMPTEL (0.8–30 MeV) and INTEGRAL/SPI (0.05–8 MeV) see the shifted photons at their spectral peak, but the measured MeV background accommodates f_PBH = 1 throughout the window. Second, extended log-normal mass functions close the window at widths σ ≳ 0.5 dex, but it survives for narrow distributions (σ ≲ 0.3 dex) characteristic of cosmic string or phase-transition formation mechanisms. Constraints on δ from BBN and inflation (δ ≲ 10⁻⁴) bind the cosmological horizon via modified Friedmann equations, not necessarily black hole horizons. The exponential sensitivity to temperature is model-independent: any entropy modification, Barrow, Tsallis, Rényi, or the generalized uncertainty principle—that cools PBH-scale black holes by a factor of a few triggers the same constraint collapse, a structural fragility of the Planck function that current exclusion plots do not reflect. The specific Barrow result requires δ_BH ≳ 0.02, currently unconstrained observationally.